$$\frac{\cos(2x)+\cos(2y)}{\sin(x)+\cos(y)} = 2\cos(y)-2\sin(x)$$
The question asks to prove the identity. I tried simplifying the first half, thought maybe I could expand and simplify with the double angle formulas.
Changed it to $$\cos(x)^2 – sin(x)^2 + cos(y)^2 – sin(y)^2$$ and tried a few thing like that, but I'm stuck at that point. Am I even on the right track here, or way off?
Best Answer
We have, using the identity $\cos^2{x}+\sin^2{x}=1$: $$ \cos{2x} = \cos^2{x}-\sin^2{x} = 2\cos^2{x}-1 = 1-2\sin^2{x}. $$ Then $$ \cos{2x}+\cos{2y} =1-2\sin^2{x} + 2\cos^2{y}-1 = 2(\cos^2{y}-\sin^2{x}) \\ = 2(\cos{y}-\sin{x})(\cos{y}+\sin{x}), $$ and dividing gives the required identity.